Preprocessing Graph Problems When Does a Small Vertex Cover Help? Bart M. P. Jansen Joint work with Fedor V. Fomin & Michał Pilipczuk June 2012, Dagstuhl Seminar 12241

Motivation Graph structure affects problem complexity Algorithmic properties of such connections are pretty well- understood: – Courcelle's Theorem – Many other approaches for parameter vertex cover What about kernelization complexity? – Many problems admit polynomial kernels – Many problems do not admit polynomial kernels 2 Which graph problems can be effectively preprocessed when the input has a small vertex cover?

Hierarchy of parameters 3

Problem setting C LIQUE PARAMETERIZED BY V ERTEX C OVER Input: A graph G, a vertex cover X of G, integer k Parameter: |X|. Question: Does G have a clique on k vertices? V ERTEX C OVER PARAMETERIZED BY V ERTEX C OVER Input: A graph G, a vertex cover X of G, integer k Parameter: |X|. Question: Does G have a vertex cover of size at most k? A vertex cover is given in the input for technical reasons – May compute a 2-approximate vertex cover for X 4 X

5 C LIQUE V ERTEX C OVER T REEWIDTH C UTWIDTH O DD C YCLE T RANSVERSAL C HROMATIC NUMBER L ONGEST P ATH q-C OLORING  -T RANSVERSAL D OMINATING S ET S TEINER T REE D ISJOINT P ATHS D ISJOINT C YCLES W EIGHTED T REEWIDTH W EIGHTED F EEDBACK V ERTEX S ET I NDEPENDENT S ET Kernelization Complexity of Parameterizations by Vertex Cover

Our results Sufficient conditions for vertex-deletion and induced subgraph problems to admit polynomial kernels Unifies many known kernels & provides new results General positive results Testing for an Ht induced subgraph / minor (Cliques, stars, bicliques, paths, cycles …) Subgraph vs. minor tests often behave differently L ONGEST I NDUCED P ATH, M AXIMUM I NDUCED M ATCHING, and I NDUCED K s,t S UBGRAPH T EST parameterized by vertex cover, have no polynomial kernel (unless NP ⊆ coNP/poly) Upper and lower bounds for subgraph and minor tests 6

D ELETION D ISTANCE TO  - FREE Sufficient conditions for polynomial kernels 7

General positive results Not about expressibility in logic Revolves around a closure property of graph families 8  Problem {K 2 }Vertex Cover Cyclic graphsFeedback Vertex Set Graphs with an odd cycleOdd Cycle Transversal Graphs with a chordless cycleChordal Deletion Graphs with a K 3,3 or K 5 minorVertex Planarization

Properties characterized by few adjacencies Graph property  is characterized by c  adjacencies if: – for any graph G in  and vertex v in G, – there is a set D ⊆ V(G) \ {v} of ≤ c  vertices, – such that all graphs G’ made from G by changing the presence of edges between v and V(G) \ D, – are contained in . 9 Example: property of having a chordless cycle (c  =3) Non-example: having an odd hole

Some properties characterized by few adjacencies Having a chordless cycle of length at least l [c = l - 1] For a Hamiltonian graph and vertex v, let D be the predecessor and successor on some Hamiltonian cycle Hamiltonicity [ c = 2 ] Let D be the neighbors of v in a minimal minor model [ deg(v) ≤  (H) ] Containing H as a minor [ c =  (H) ]Any finite set of graphs [ c = maxH |V(H)| - 1  10 ( ∪ ’) is characterized by max(c , c ’ ) adjacencies ( ∩ ’) is characterized by c  +c ’ adjacencies

Generic kernelization scheme for D ELETION D ISTANCE TO  -F REE 11 Deletion Distance to {2 · K 1 }-Free is C LIQUE, for which a lower bound exists Set of forbidden graphs behaves “nicely” All forbidden graphs contain an induced subgraph of size polynomial in their VC number For C HORDAL D ELETION let  be graphs with a chordless cycle i.Characterized by 3 adjacencies ii.All graphs with a chordless cycle have ≥ 4 edges iii.Satisfied for p(x) = 2x Vertex-minimal graphs with a chordless cycle are Hamiltonian For Hamiltonian graphs G it holds that |V(G)| ≤ 2 VC (G) C HORDAL D ELETION has a kernel with O( (x + 2x) · x 3 ) = O(x 4 ) vertices

Reduction rule R EDUCE (Graph G, Vertex cover X, integer l, integer c  ) For each Y ⊆ X of size at most c  – For each partition of Y into Y + and Y - Let Z be the vertices in V(G) \ X adjacent to all of Y + and none of Y - Mark l arbitrary vertices from Z Delete all unmarked vertices not in X 12 X Reduce(G, X, l, c) results in a graph on O(|X| + l · c · 2 c · |X| c ) vertices Example for c = 3 and l = 2

Kernelization strategy 13 Kernelization for input (G, X, k) If k ≥ |X| then output YES – Condition (ii): all forbidden graphs in  have at least one edge, so X is a solution of size ≤ k Return R EDUCE (G, X, k + p(|X|), c  ) Size bound follows immediately from reduction rule

Correctness (I) Suppose (G,X,k) is transformed into (G’,X,k) G’ is an induced subgraph of G – G-S is -free implies that G’-S is -free Reverse direction: any solution S in G’ is a solution in G – Proof… 14

Correctness (II) G’-S  -free  G-S  -free Reduction deletes some unmarked vertices Z Add vertices from Z back to G’-S to build G-S If adding v creates some forbidden graph H from , consider the set D such that changing adjacencies between v and V(H)\D in H, preserves membership in  – We marked k + p(|X|) vertices that see exactly the same as v in D ∩ X – |S| ≤ k and |V(H)| ≤ p(|X|) by Condition (iii) – There is some marked vertex u, not in H, that sees the same as v in D ∩ X As u and v do not belong to the vertex cover, neither sees any vertices outside X – u and v see the same in D \ X, and hence u and v see the same in D Replace v by u in H, to get some H’ – H’ can be made from H by changing edges between v and V(H) \ D – So H’ is forbidden (condition (i)) – contradiction 15 v d1d1 d2d2 d3d3 u X

Implications of the theorem Polynomial kernels for the following problems parameterized by the size x of a given vertex cover 16 V ERTEX C OVER O(x 2 ) vertices O DD C YCLE T RANSVERSAL O(x 3 ) vertices F EEDBACK V ERTEX S ET O(x 3 ) vertices C HORDAL V ERTEX D ELETION O(x 4 ) vertices V ERTEX P LANARIZATION O(x 5 ) vertices  -T RANSVERSAL O(x f(  ) ) vertices  -M INOR -F REE D ELETION O(x  +1 ) D ISTANCE H EREDITARY V ERTEX D ELETION O(X 6 ) C HORDAL B IPARTITE V ERTEX D ELETION O(X 5 ) P ATHWIDTH -t V ERTEX D ELETION O(x f(t) ) vertices

L ARGEST I NDUCED  - SUBGRAPH Sufficient conditions for polynomial kernels 17

General positive results 18  Problem Hamiltonian graphsL ONGEST C YCLE Graphs with a Hamiltonian pathL ONGEST P ATH Graphs partitionable into trianglesT RIANGLE P ACKING Graphs partitionable into vertex-disjoint HH-P ACKING

MINOR TESTING VS. INDUCED SUBGRAPH TESTING 19

Kernelization complexity overview Graph familyInduced subgraph testingMinor testing Cliques K t No polynomial kernelPolynomial kernel * Stars K 1,t Polynomial kernel *No polynomial kernel Bicliques K s,t No polynomial kernel *No polynomial kernel Paths P t No polynomial kernel *Polynomial kernel Matchings t · K 2 No polynomial kernel *P-time solvable 20 Problems are parameterized by the size of a given VC Size t of the tested graph is part of the input

Conclusion Generic reduction scheme yields polynomial kernels for D ELETION D ISTANCE TO  - FREE and L ARGEST I NDUCED  - SUBGRAPH Gives insight into why polynomial kernels exist for these cases – Expressibility with respect to forbidden / desired graph properties  that are characterized by few adjacencies Differing kernelization complexity of minor vs. induced subgraph testing Open problems: – Are there polynomial kernels for P ERFECT V ERTEX D ELETION B ANDWIDTH parameterized by Vertex Cover? – More general theorems that also capture T REEWIDTH, C LIQUE M INOR T EST, etc.? 21